Axiom ax-mulass 7509

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7469. Proofs should normally use mulass 7534 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7409	. . . 4 class ℂ
3	1, 2	wcel 1439	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1439	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1439	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 925	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7416	. . . . 5 class ·
10	1, 4, 9	co 5666	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5666	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5666	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5666	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1290	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7534